Geometric Distribution Calculator
Calculate geometric probability ($P(X = k) = (1-p)^{k-1} p$) for first success occurring on trial $k$, expected trials ($E(X) = 1/p$), and variance.
Calculate Geometric Trial Probabilities
Enter success probability (p) and trial number of 1st success (k).
Calculation Results
Calculated using geometric distribution probability mass function: P(X = k) = (1 - p)^{k - 1} \cdot p
Quick Summary
The Geometric Distribution Calculator evaluates exact probability $P(X = k)$, cumulative probabilities $P(X \le k)$, expected trials to first success ($1/p$), and variance.
Formula Explanation
P(X = k) = (1 - p)^{k - 1} \cdot p
P(X \le k) = 1 - (1 - p)^k
E(X) = \frac{1}{p}, \quad \text{Var}(X) = \frac{1 - p}{p^2}How It Works
The geometric distribution models the number of trials needed to achieve the first success in repeated independent Bernoulli trials. The Geometric Distribution Calculator computes exact probability $P(X=k)$, cumulative distribution $P(X \le k)$, and expected waiting trials $E(X) = 1/p$.
Step-by-Step Worked Example
Practical Problem: A basketball player makes free throws with $p = 0.20$ (20% success rate). Calculate probability that the first successful shot occurs on trial $k = 3$.
- Step 1: Calculate failure probability ($1 - p$): $1 - 0.20 = \mathbf{0.80}$.
- Step 2: Calculate failure factor for first $k - 1 = 2$ trials: $0.80^2 = \mathbf{0.64}$.
- Step 3: Multiply by success probability on 3rd trial ($p = 0.20$): $P(X = 3) = 0.64 \times 0.20 = \mathbf{0.1280\text{ (12.80\%)}}.$
- Step 4: Calculate Cumulative $P(X \le 3)$: $1 - 0.80^3 = 1 - 0.512 = \mathbf{0.4880\text{ (48.80\%)}}.$
- Step 5: Calculate Expected Trials $E(X)$: $E(X) = 1 / 0.20 = \mathbf{5.00\text{ trials required on average}}$.
Real-World Calculation Examples
Scenario 1: Basketball Free Throws (p = 0.20, k = 3)
Parameters: p = 0.20, k = 3
Result: P(X = 3) = 0.1280 (Expected trials E(X) = 5.00).
Scenario 2: Rolling a 6 on a Die (p = 1/6)
Parameters: p = 0.1667, k = 1
Result: P(X = 1) = 0.1667 (Expected trials E(X) = 6.00).
Scenario 3: Machine Component Failure Inspection
Parameters: p = 0.05 defect rate, k ≤ 10
Result: P(X ≤ 10) = 0.4013.
Scenario 4: Sales Prospecting First Conversion
Parameters: p = 0.10 conversion, k = 1st call
Result: Expected calls E(X) = 10 calls.
Key Benefits of Using This Calculator
Expected Waiting Trials ($1/p$)
Calculates average expected trials needed until first success.
Survival & Cumulative Probabilities
Outputs $P(X \le k)$ and survival probability $P(X > k) = (1-p)^k$.
Memoryless Property Modeling
Accurately models memoryless discrete trial sequences.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a geometric distribution?
The geometric distribution models the probability of needing $k$ independent Bernoulli trials to get the first success.
What is the memoryless property of geometric distribution?
The memoryless property means past unsuccessful trials do not affect future trial probabilities ($P(X > s + t \mid X > s) = P(X > t)$).
What is formula for expected value and variance?
$E(X) = \frac{1}{p}$ and $\text{Var}(X) = \frac{1 - p}{p^2}$.
How do I calculate geometric probabilities in Excel?
Use formula =NEGBINOM.DIST(k - 1, 1, p, FALSE) for exact $P(X=k)$.
What is difference between geometric and negative binomial distribution?
Geometric distribution models trials for the 1st success; negative binomial models trials for $r$-th success ($r \ge 1$).
What is continuous equivalent of geometric distribution?
The Exponential distribution is the continuous memoryless counterpart of the discrete geometric distribution.
Can trial k be 0?
In this formulation (number of trials), $k \ge 1$. In alternative formulation (number of failures before 1st success), $k \ge 0$.
What is survival function $P(X > k)$?
Survival function $P(X > k) = (1 - p)^k$ is the probability that the first $k$ trials all fail.
What is median of a geometric distribution?
$\text{Median} = \left\lceil \frac{-\ln 2}{\ln(1 - p)} \right\rceil$.
Why is geometric mean different from geometric distribution?
Geometric mean is a central tendency average ($\sqrt[n]{\prod x_i}$); geometric distribution is a discrete probability distribution for first success trials.